LMFDB - Embedded newform 1850.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.53175$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} +1.53175 q^{12} +5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} +3.60861 q^{17} -0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} +0.0451320 q^{22} -2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} -5.59662 q^{27} +4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} +1.00000 q^{32} +0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} +1.00000 q^{37} -6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} +7.15650 q^{42} -5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} -4.28072 q^{47} +1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} +5.26514 q^{52} +6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} -9.52749 q^{57} -4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} -3.01051 q^{62} -3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} +12.3751 q^{67} +3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} -0.653743 q^{72} +7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} +0.210861 q^{77} +8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} -7.38299 q^{82} -0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} -6.43926 q^{87} +0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} -2.20164 q^{92} -4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} +0.240408 q^{97} +14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9	$ 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 + 3 * q^11 - 6 * q^13 - q^14 + 5 * q^16 + 9 * q^17 + 3 * q^18 + 4 * q^19 + 16 * q^21 + 3 * q^22 + 6 * q^23 - 6 * q^26 - q^28 + 11 * q^29 + 23 * q^31 + 5 * q^32 + 20 * q^33 + 9 * q^34 + 3 * q^36 + 5 * q^37 + 4 * q^38 + 20 * q^39 - 7 * q^41 + 16 * q^42 - 17 * q^43 + 3 * q^44 + 6 * q^46 + 12 * q^47 + 30 * q^49 - 20 * q^51 - 6 * q^52 - 7 * q^53 - q^56 + 11 * q^58 + 20 * q^59 - 9 * q^61 + 23 * q^62 - 33 * q^63 + 5 * q^64 + 20 * q^66 + 12 * q^67 + 9 * q^68 + 16 * q^69 + 6 * q^71 + 3 * q^72 - 6 * q^73 + 5 * q^74 + 4 * q^76 - q^77 + 20 * q^78 + 20 * q^79 - 7 * q^81 - 7 * q^82 + 12 * q^83 + 16 * q^84 - 17 * q^86 - 34 * q^87 + 3 * q^88 + 12 * q^89 + 16 * q^91 + 6 * q^92 - 4 * q^93 + 12 * q^94 + 3 * q^97 + 30 * q^98 - 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.53175	0.884356	0.442178	−	0.896927i	$-0.354206\pi$
$3$	1.53175	0.884356	0.442178	+	0.896927i	$0.354206\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.53175	0.625334
$7$	4.67211	1.76589	0.882946	−	0.469475i	$-0.155557\pi$
$7$	4.67211	1.76589	0.882946	+	0.469475i	$0.155557\pi$
$8$	1.00000	0.353553
$9$	−0.653743	−0.217914
$10$	0	0
$11$	0.0451320	0.0136078	0.00680390	−	0.999977i	$-0.497834\pi$
$11$	0.0451320	0.0136078	0.00680390	+	0.999977i	$0.497834\pi$
$12$	1.53175	0.442178
$13$	5.26514	1.46029	0.730143	−	0.683294i	$-0.239453\pi$
$13$	5.26514	1.46029	0.730143	+	0.683294i	$0.239453\pi$
$14$	4.67211	1.24867
$15$	0	0
$16$	1.00000	0.250000
$17$	3.60861	0.875217	0.437608	−	0.899166i	$-0.355826\pi$
$17$	3.60861	0.875217	0.437608	+	0.899166i	$0.355826\pi$
$18$	−0.653743	−0.154089
$19$	−6.22000	−1.42697	−0.713483	−	0.700672i	$-0.752884\pi$
$19$	−6.22000	−1.42697	−0.713483	+	0.700672i	$0.752884\pi$
$20$	0	0
$21$	7.15650	1.56168
$22$	0.0451320	0.00962217
$23$	−2.20164	−0.459073	−0.229536	−	0.973300i	$-0.573721\pi$
$23$	−2.20164	−0.459073	−0.229536	+	0.973300i	$0.573721\pi$
$24$	1.53175	0.312667
$25$	0	0
$26$	5.26514	1.03258
$27$	−5.59662	−1.07707
$28$	4.67211	0.882946
$29$	−4.20386	−0.780637	−0.390319	−	0.920680i	$-0.627635\pi$
$29$	−4.20386	−0.780637	−0.390319	+	0.920680i	$0.627635\pi$
$30$	0	0
$31$	−3.01051	−0.540704	−0.270352	−	0.962762i	$-0.587140\pi$
$31$	−3.01051	−0.540704	−0.270352	+	0.962762i	$0.587140\pi$
$32$	1.00000	0.176777
$33$	0.0691309	0.0120341
$34$	3.60861	0.618872
$35$	0	0
$36$	−0.653743	−0.108957
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−6.22000	−1.00902
$39$	8.06487	1.29141
$40$	0	0
$41$	−7.38299	−1.15303	−0.576515	−	0.817087i	$-0.695588\pi$
$41$	−7.38299	−1.15303	−0.576515	+	0.817087i	$0.695588\pi$
$42$	7.15650	1.10427
$43$	−5.54789	−0.846046	−0.423023	−	0.906119i	$-0.639031\pi$
$43$	−5.54789	−0.846046	−0.423023	+	0.906119i	$0.639031\pi$
$44$	0.0451320	0.00680390
$45$	0	0
$46$	−2.20164	−0.324613
$47$	−4.28072	−0.624407	−0.312204	−	0.950015i	$-0.601067\pi$
$47$	−4.28072	−0.624407	−0.312204	+	0.950015i	$0.601067\pi$
$48$	1.53175	0.221089
$49$	14.8286	2.11837
$50$	0	0
$51$	5.52749	0.774003
$52$	5.26514	0.730143
$53$	6.10215	0.838194	0.419097	−	0.907941i	$-0.362347\pi$
$53$	6.10215	0.838194	0.419097	+	0.907941i	$0.362347\pi$
$54$	−5.59662	−0.761603
$55$	0	0
$56$	4.67211	0.624337
$57$	−9.52749	−1.26195
$58$	−4.20386	−0.551994
$59$	13.5275	1.76113	0.880565	−	0.473926i	$-0.157164\pi$
$59$	13.5275	1.76113	0.880565	+	0.473926i	$0.157164\pi$
$60$	0	0
$61$	−4.27299	−0.547100	−0.273550	−	0.961858i	$-0.588198\pi$
$61$	−4.27299	−0.547100	−0.273550	+	0.961858i	$0.588198\pi$
$62$	−3.01051	−0.382335
$63$	−3.05436	−0.384813
$64$	1.00000	0.125000
$65$	0	0
$66$	0.0691309	0.00850942
$67$	12.3751	1.51186	0.755932	−	0.654650i	$-0.227184\pi$
$67$	12.3751	1.51186	0.755932	+	0.654650i	$0.227184\pi$
$68$	3.60861	0.437608
$69$	−3.37235	−0.405984
$70$	0	0
$71$	−4.49354	−0.533285	−0.266642	−	0.963796i	$-0.585914\pi$
$71$	−4.49354	−0.533285	−0.266642	+	0.963796i	$0.585914\pi$
$72$	−0.653743	−0.0770443
$73$	7.03811	0.823748	0.411874	−	0.911241i	$-0.364874\pi$
$73$	7.03811	0.823748	0.411874	+	0.911241i	$0.364874\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−6.22000	−0.713483
$77$	0.210861	0.0240299
$78$	8.06487	0.913167
$79$	−8.72499	−0.981638	−0.490819	−	0.871262i	$-0.663302\pi$
$79$	−8.72499	−0.981638	−0.490819	+	0.871262i	$0.663302\pi$
$80$	0	0
$81$	−6.61139	−0.734599
$82$	−7.38299	−0.815315
$83$	−0.880231	−0.0966179	−0.0483090	−	0.998832i	$-0.515383\pi$
$83$	−0.880231	−0.0966179	−0.0483090	+	0.998832i	$0.515383\pi$
$84$	7.15650	0.780839
$85$	0	0
$86$	−5.54789	−0.598245
$87$	−6.43926	−0.690361
$88$	0.0451320	0.00481108
$89$	9.97602	1.05746	0.528728	−	0.848791i	$-0.322669\pi$
$89$	9.97602	1.05746	0.528728	+	0.848791i	$0.322669\pi$
$90$	0	0
$91$	24.5993	2.57871
$92$	−2.20164	−0.229536
$93$	−4.61135	−0.478175
$94$	−4.28072	−0.441523
$95$	0	0
$96$	1.53175	0.156334
$97$	0.240408	0.0244097	0.0122049	−	0.999926i	$-0.496115\pi$
$97$	0.240408	0.0244097	0.0122049	+	0.999926i	$0.496115\pi$
$98$	14.8286	1.49792
$99$	−0.0295047	−0.00296533
$100$	0	0
$101$	−5.61775	−0.558987	−0.279494	−	0.960148i	$-0.590167\pi$
$101$	−5.61775	−0.558987	−0.279494	+	0.960148i	$0.590167\pi$
$102$	5.52749	0.547303
$103$	−12.5663	−1.23819	−0.619095	−	0.785316i	$-0.712501\pi$
$103$	−12.5663	−1.23819	−0.619095	+	0.785316i	$0.712501\pi$
$104$	5.26514	0.516289
$105$	0	0
$106$	6.10215	0.592693
$107$	−17.9395	−1.73427	−0.867137	−	0.498070i	$-0.834042\pi$
$107$	−17.9395	−1.73427	−0.867137	+	0.498070i	$0.834042\pi$
$108$	−5.59662	−0.538535
$109$	3.09936	0.296865	0.148433	−	0.988923i	$-0.452577\pi$
$109$	3.09936	0.296865	0.148433	+	0.988923i	$0.452577\pi$
$110$	0	0
$111$	1.53175	0.145387
$112$	4.67211	0.441473
$113$	4.36184	0.410328	0.205164	−	0.978728i	$-0.434227\pi$
$113$	4.36184	0.410328	0.205164	+	0.978728i	$0.434227\pi$
$114$	−9.52749	−0.892331
$115$	0	0
$116$	−4.20386	−0.390319
$117$	−3.44204	−0.318217
$118$	13.5275	1.24531
$119$	16.8598	1.54554
$120$	0	0
$121$	−10.9980	−0.999815
$122$	−4.27299	−0.386858
$123$	−11.3089	−1.01969
$124$	−3.01051	−0.270352
$125$	0	0
$126$	−3.05436	−0.272104
$127$	17.6572	1.56683	0.783413	−	0.621502i	$-0.213477\pi$
$127$	17.6572	1.56683	0.783413	+	0.621502i	$0.213477\pi$
$128$	1.00000	0.0883883
$129$	−8.49798	−0.748206
$130$	0	0
$131$	−10.3103	−0.900812	−0.450406	−	0.892824i	$-0.648721\pi$
$131$	−10.3103	−0.900812	−0.450406	+	0.892824i	$0.648721\pi$
$132$	0.0691309	0.00601707
$133$	−29.0605	−2.51987
$134$	12.3751	1.06905
$135$	0	0
$136$	3.60861	0.309436
$137$	3.21863	0.274986	0.137493	−	0.990503i	$-0.456095\pi$
$137$	3.21863	0.274986	0.137493	+	0.990503i	$0.456095\pi$
$138$	−3.37235	−0.287074
$139$	20.9400	1.77611	0.888055	−	0.459737i	$-0.152056\pi$
$139$	20.9400	1.77611	0.888055	+	0.459737i	$0.152056\pi$
$140$	0	0
$141$	−6.55699	−0.552198
$142$	−4.49354	−0.377089
$143$	0.237626	0.0198713
$144$	−0.653743	−0.0544786
$145$	0	0
$146$	7.03811	0.582478
$147$	22.7137	1.87340
$148$	1.00000	0.0821995
$149$	−21.9832	−1.80093	−0.900467	−	0.434924i	$-0.856775\pi$
$149$	−21.9832	−1.80093	−0.900467	+	0.434924i	$0.856775\pi$
$150$	0	0
$151$	14.1877	1.15458	0.577290	−	0.816539i	$-0.304110\pi$
$151$	14.1877	1.15458	0.577290	+	0.816539i	$0.304110\pi$
$152$	−6.22000	−0.504509
$153$	−2.35910	−0.190722
$154$	0.210861	0.0169917
$155$	0	0
$156$	8.06487	0.645706
$157$	−19.9316	−1.59072	−0.795359	−	0.606139i	$-0.792717\pi$
$157$	−19.9316	−1.59072	−0.795359	+	0.606139i	$0.792717\pi$
$158$	−8.72499	−0.694123
$159$	9.34696	0.741262
$160$	0	0
$161$	−10.2863	−0.810673
$162$	−6.61139	−0.519440
$163$	8.65087	0.677588	0.338794	−	0.940861i	$-0.389981\pi$
$163$	8.65087	0.677588	0.338794	+	0.940861i	$0.389981\pi$
$164$	−7.38299	−0.576515
$165$	0	0
$166$	−0.880231	−0.0683192
$167$	−11.9640	−0.925803	−0.462901	−	0.886410i	$-0.653192\pi$
$167$	−11.9640	−0.925803	−0.462901	+	0.886410i	$0.653192\pi$
$168$	7.15650	0.552136
$169$	14.7216	1.13243
$170$	0	0
$171$	4.06628	0.310956
$172$	−5.54789	−0.423023
$173$	−5.33508	−0.405619	−0.202809	−	0.979218i	$-0.565007\pi$
$173$	−5.33508	−0.405619	−0.202809	+	0.979218i	$0.565007\pi$
$174$	−6.43926	−0.488159
$175$	0	0
$176$	0.0451320	0.00340195
$177$	20.7207	1.55747
$178$	9.97602	0.747734
$179$	−5.27905	−0.394575	−0.197288	−	0.980346i	$-0.563213\pi$
$179$	−5.27905	−0.394575	−0.197288	+	0.980346i	$0.563213\pi$
$180$	0	0
$181$	17.7448	1.31896	0.659478	−	0.751723i	$-0.270777\pi$
$181$	17.7448	1.31896	0.659478	+	0.751723i	$0.270777\pi$
$182$	24.5993	1.82342
$183$	−6.54515	−0.483832
$184$	−2.20164	−0.162307
$185$	0	0
$186$	−4.61135	−0.338121
$187$	0.162864	0.0119098
$188$	−4.28072	−0.312204
$189$	−26.1480	−1.90199
$190$	0	0
$191$	−3.11649	−0.225501	−0.112751	−	0.993623i	$-0.535966\pi$
$191$	−3.11649	−0.225501	−0.112751	+	0.993623i	$0.535966\pi$
$192$	1.53175	0.110545
$193$	−8.78278	−0.632198	−0.316099	−	0.948726i	$-0.602373\pi$
$193$	−8.78278	−0.632198	−0.316099	+	0.948726i	$0.602373\pi$
$194$	0.240408	0.0172603
$195$	0	0
$196$	14.8286	1.05919
$197$	7.00278	0.498928	0.249464	−	0.968384i	$-0.419746\pi$
$197$	7.00278	0.498928	0.249464	+	0.968384i	$0.419746\pi$
$198$	−0.0295047	−0.00209681
$199$	17.3202	1.22780	0.613900	−	0.789384i	$-0.289600\pi$
$199$	17.3202	1.22780	0.613900	+	0.789384i	$0.289600\pi$
$200$	0	0
$201$	18.9556	1.33703
$202$	−5.61775	−0.395264
$203$	−19.6409	−1.37852
$204$	5.52749	0.387002
$205$	0	0
$206$	−12.5663	−0.875533
$207$	1.43930	0.100038
$208$	5.26514	0.365071
$209$	−0.280721	−0.0194179
$210$	0	0
$211$	6.77438	0.466368	0.233184	−	0.972433i	$-0.425086\pi$
$211$	6.77438	0.466368	0.233184	+	0.972433i	$0.425086\pi$
$212$	6.10215	0.419097
$213$	−6.88297	−0.471613
$214$	−17.9395	−1.22632
$215$	0	0
$216$	−5.59662	−0.380802
$217$	−14.0654	−0.954825
$218$	3.09936	0.209915
$219$	10.7806	0.728487
$220$	0	0
$221$	18.9998	1.27807
$222$	1.53175	0.102804
$223$	−10.2559	−0.686784	−0.343392	−	0.939192i	$-0.611576\pi$
$223$	−10.2559	−0.686784	−0.343392	+	0.939192i	$0.611576\pi$
$224$	4.67211	0.312168
$225$	0	0
$226$	4.36184	0.290145
$227$	−11.2744	−0.748306	−0.374153	−	0.927367i	$-0.622066\pi$
$227$	−11.2744	−0.748306	−0.374153	+	0.927367i	$0.622066\pi$
$228$	−9.52749	−0.630973
$229$	−9.40494	−0.621496	−0.310748	−	0.950492i	$-0.600579\pi$
$229$	−9.40494	−0.621496	−0.310748	+	0.950492i	$0.600579\pi$
$230$	0	0
$231$	0.322987	0.0212510
$232$	−4.20386	−0.275997
$233$	−3.22867	−0.211517	−0.105759	−	0.994392i	$-0.533727\pi$
$233$	−3.22867	−0.211517	−0.105759	+	0.994392i	$0.533727\pi$
$234$	−3.44204	−0.225013
$235$	0	0
$236$	13.5275	0.880565
$237$	−13.3645	−0.868118
$238$	16.8598	1.09286
$239$	−17.8612	−1.15534	−0.577672	−	0.816269i	$-0.696039\pi$
$239$	−17.8612	−1.15534	−0.577672	+	0.816269i	$0.696039\pi$
$240$	0	0
$241$	26.6840	1.71887	0.859434	−	0.511248i	$-0.170816\pi$
$241$	26.6840	1.71887	0.859434	+	0.511248i	$0.170816\pi$
$242$	−10.9980	−0.706976
$243$	6.66286	0.427423
$244$	−4.27299	−0.273550
$245$	0	0
$246$	−11.3089	−0.721029
$247$	−32.7492	−2.08378
$248$	−3.01051	−0.191168
$249$	−1.34829	−0.0854447
$250$	0	0
$251$	−30.1552	−1.90338	−0.951690	−	0.307060i	$-0.900655\pi$
$251$	−30.1552	−1.90338	−0.951690	+	0.307060i	$0.900655\pi$
$252$	−3.05436	−0.192406
$253$	−0.0993641	−0.00624697
$254$	17.6572	1.10791
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.15376	−0.196726	−0.0983632	−	0.995151i	$-0.531361\pi$
$257$	−3.15376	−0.196726	−0.0983632	+	0.995151i	$0.531361\pi$
$258$	−8.49798	−0.529061
$259$	4.67211	0.290311
$260$	0	0
$261$	2.74824	0.170112
$262$	−10.3103	−0.636970
$263$	14.0332	0.865322	0.432661	−	0.901557i	$-0.357575\pi$
$263$	14.0332	0.865322	0.432661	+	0.901557i	$0.357575\pi$
$264$	0.0691309	0.00425471
$265$	0	0
$266$	−29.0605	−1.78182
$267$	15.2808	0.935167
$268$	12.3751	0.755932
$269$	−30.1257	−1.83679	−0.918397	−	0.395660i	$-0.870516\pi$
$269$	−30.1257	−1.83679	−0.918397	+	0.395660i	$0.870516\pi$
$270$	0	0
$271$	14.8802	0.903910	0.451955	−	0.892041i	$-0.350727\pi$
$271$	14.8802	0.903910	0.451955	+	0.892041i	$0.350727\pi$
$272$	3.60861	0.218804
$273$	37.6800	2.28049
$274$	3.21863	0.194445
$275$	0	0
$276$	−3.37235	−0.202992
$277$	0.147278	0.00884908	0.00442454	−	0.999990i	$-0.498592\pi$
$277$	0.147278	0.00884908	0.00442454	+	0.999990i	$0.498592\pi$
$278$	20.9400	1.25590
$279$	1.96810	0.117827
$280$	0	0
$281$	−25.5247	−1.52268	−0.761339	−	0.648354i	$-0.775458\pi$
$281$	−25.5247	−1.52268	−0.761339	+	0.648354i	$0.775458\pi$
$282$	−6.55699	−0.390463
$283$	0.0562691	0.00334485	0.00167242	−	0.999999i	$-0.499468\pi$
$283$	0.0562691	0.00334485	0.00167242	+	0.999999i	$0.499468\pi$
$284$	−4.49354	−0.266642
$285$	0	0
$286$	0.237626	0.0140511
$287$	−34.4942	−2.03613
$288$	−0.653743	−0.0385222
$289$	−3.97793	−0.233996
$290$	0	0
$291$	0.368245	0.0215869
$292$	7.03811	0.411874
$293$	13.7033	0.800557	0.400278	−	0.916394i	$-0.368913\pi$
$293$	13.7033	0.800557	0.400278	+	0.916394i	$0.368913\pi$
$294$	22.7137	1.32469
$295$	0	0
$296$	1.00000	0.0581238
$297$	−0.252586	−0.0146565
$298$	−21.9832	−1.27345
$299$	−11.5919	−0.670377
$300$	0	0
$301$	−25.9204	−1.49403
$302$	14.1877	0.816411
$303$	−8.60499	−0.494344
$304$	−6.22000	−0.356742
$305$	0	0
$306$	−2.35910	−0.134861
$307$	−17.5823	−1.00348	−0.501738	−	0.865020i	$-0.667306\pi$
$307$	−17.5823	−1.00348	−0.501738	+	0.865020i	$0.667306\pi$
$308$	0.210861	0.0120149
$309$	−19.2484	−1.09500
$310$	0	0
$311$	0.353133	0.0200244	0.0100122	−	0.999950i	$-0.496813\pi$
$311$	0.353133	0.0200244	0.0100122	+	0.999950i	$0.496813\pi$
$312$	8.06487	0.456583
$313$	−18.8717	−1.06669	−0.533346	−	0.845897i	$-0.679066\pi$
$313$	−18.8717	−1.06669	−0.533346	+	0.845897i	$0.679066\pi$
$314$	−19.9316	−1.12481
$315$	0	0
$316$	−8.72499	−0.490819
$317$	15.1979	0.853601	0.426800	−	0.904346i	$-0.359641\pi$
$317$	15.1979	0.853601	0.426800	+	0.904346i	$0.359641\pi$
$318$	9.34696	0.524152
$319$	−0.189728	−0.0106228
$320$	0	0
$321$	−27.4788	−1.53372
$322$	−10.2863	−0.573232
$323$	−22.4456	−1.24890
$324$	−6.61139	−0.367300
$325$	0	0
$326$	8.65087	0.479127
$327$	4.74745	0.262535
$328$	−7.38299	−0.407658
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−23.0210	−1.26535	−0.632674	−	0.774418i	$-0.718043\pi$
$331$	−23.0210	−1.26535	−0.632674	+	0.774418i	$0.718043\pi$
$332$	−0.880231	−0.0483090
$333$	−0.653743	−0.0358249
$334$	−11.9640	−0.654641
$335$	0	0
$336$	7.15650	0.390419
$337$	34.0776	1.85633	0.928163	−	0.372173i	$-0.121387\pi$
$337$	34.0776	1.85633	0.928163	+	0.372173i	$0.121387\pi$
$338$	14.7216	0.800752
$339$	6.68125	0.362876
$340$	0	0
$341$	−0.135870	−0.00735779
$342$	4.06628	0.219879
$343$	36.5761	1.97493
$344$	−5.54789	−0.299122
$345$	0	0
$346$	−5.33508	−0.286816
$347$	13.8449	0.743236	0.371618	−	0.928386i	$-0.378803\pi$
$347$	13.8449	0.743236	0.371618	+	0.928386i	$0.378803\pi$
$348$	−6.43926	−0.345181
$349$	11.3765	0.608970	0.304485	−	0.952517i	$-0.401516\pi$
$349$	11.3765	0.608970	0.304485	+	0.952517i	$0.401516\pi$
$350$	0	0
$351$	−29.4670	−1.57283
$352$	0.0451320	0.00240554
$353$	19.2631	1.02527	0.512636	−	0.858606i	$-0.328669\pi$
$353$	19.2631	1.02527	0.512636	+	0.858606i	$0.328669\pi$
$354$	20.7207	1.10129
$355$	0	0
$356$	9.97602	0.528728
$357$	25.8250	1.36681
$358$	−5.27905	−0.279007
$359$	14.6645	0.773961	0.386980	−	0.922088i	$-0.373518\pi$
$359$	14.6645	0.773961	0.386980	+	0.922088i	$0.373518\pi$
$360$	0	0
$361$	19.6884	1.03623
$362$	17.7448	0.932643
$363$	−16.8461	−0.884192
$364$	24.5993	1.28935
$365$	0	0
$366$	−6.54515	−0.342121
$367$	0.868349	0.0453275	0.0226637	−	0.999743i	$-0.492785\pi$
$367$	0.868349	0.0453275	0.0226637	+	0.999743i	$0.492785\pi$
$368$	−2.20164	−0.114768
$369$	4.82658	0.251262
$370$	0	0
$371$	28.5099	1.48016
$372$	−4.61135	−0.239087
$373$	32.4555	1.68048	0.840240	−	0.542214i	$-0.182414\pi$
$373$	32.4555	1.68048	0.840240	+	0.542214i	$0.182414\pi$
$374$	0.162864	0.00842148
$375$	0	0
$376$	−4.28072	−0.220761
$377$	−22.1339	−1.13995
$378$	−26.1480	−1.34491
$379$	−30.2511	−1.55389	−0.776947	−	0.629566i	$-0.783233\pi$
$379$	−30.2511	−1.55389	−0.776947	+	0.629566i	$0.783233\pi$
$380$	0	0
$381$	27.0465	1.38563
$382$	−3.11649	−0.159453
$383$	14.4935	0.740585	0.370293	−	0.928915i	$-0.379257\pi$
$383$	14.4935	0.740585	0.370293	+	0.928915i	$0.379257\pi$
$384$	1.53175	0.0781668
$385$	0	0
$386$	−8.78278	−0.447032
$387$	3.62689	0.184365
$388$	0.240408	0.0122049
$389$	36.9033	1.87107	0.935537	−	0.353229i	$-0.114916\pi$
$389$	36.9033	1.87107	0.935537	+	0.353229i	$0.114916\pi$
$390$	0	0
$391$	−7.94485	−0.401788
$392$	14.8286	0.748958
$393$	−15.7927	−0.796639
$394$	7.00278	0.352795
$395$	0	0
$396$	−0.0295047	−0.00148267
$397$	22.9365	1.15115	0.575575	−	0.817749i	$-0.304778\pi$
$397$	22.9365	1.15115	0.575575	+	0.817749i	$0.304778\pi$
$398$	17.3202	0.868185
$399$	−44.5135	−2.22846
$400$	0	0
$401$	27.4332	1.36995	0.684973	−	0.728568i	$-0.259814\pi$
$401$	27.4332	1.36995	0.684973	+	0.728568i	$0.259814\pi$
$402$	18.9556	0.945420
$403$	−15.8508	−0.789582
$404$	−5.61775	−0.279494
$405$	0	0
$406$	−19.6409	−0.974761
$407$	0.0451320	0.00223711
$408$	5.52749	0.273651
$409$	−21.3920	−1.05777	−0.528883	−	0.848695i	$-0.677389\pi$
$409$	−21.3920	−1.05777	−0.528883	+	0.848695i	$0.677389\pi$
$410$	0	0
$411$	4.93014	0.243186
$412$	−12.5663	−0.619095
$413$	63.2019	3.10996
$414$	1.43930	0.0707379
$415$	0	0
$416$	5.26514	0.258144
$417$	32.0749	1.57071
$418$	−0.280721	−0.0137305
$419$	23.0891	1.12797	0.563987	−	0.825784i	$-0.309267\pi$
$419$	23.0891	1.12797	0.563987	+	0.825784i	$0.309267\pi$
$420$	0	0
$421$	−1.83757	−0.0895576	−0.0447788	−	0.998997i	$-0.514258\pi$
$421$	−1.83757	−0.0895576	−0.0447788	+	0.998997i	$0.514258\pi$
$422$	6.77438	0.329772
$423$	2.79849	0.136067
$424$	6.10215	0.296346
$425$	0	0
$426$	−6.88297	−0.333481
$427$	−19.9639	−0.966120
$428$	−17.9395	−0.867137
$429$	0.363983	0.0175733
$430$	0	0
$431$	−15.9796	−0.769710	−0.384855	−	0.922977i	$-0.625749\pi$
$431$	−15.9796	−0.769710	−0.384855	+	0.922977i	$0.625749\pi$
$432$	−5.59662	−0.269267
$433$	−25.2525	−1.21356	−0.606779	−	0.794870i	$-0.707539\pi$
$433$	−25.2525	−1.21356	−0.606779	+	0.794870i	$0.707539\pi$
$434$	−14.0654	−0.675163
$435$	0	0
$436$	3.09936	0.148433
$437$	13.6942	0.655082
$438$	10.7806	0.515118
$439$	−9.30792	−0.444243	−0.222121	−	0.975019i	$-0.571298\pi$
$439$	−9.30792	−0.444243	−0.222121	+	0.975019i	$0.571298\pi$
$440$	0	0
$441$	−9.69410	−0.461624
$442$	18.9998	0.903729
$443$	−9.52445	−0.452520	−0.226260	−	0.974067i	$-0.572650\pi$
$443$	−9.52445	−0.452520	−0.226260	+	0.974067i	$0.572650\pi$
$444$	1.53175	0.0726936
$445$	0	0
$446$	−10.2559	−0.485629
$447$	−33.6728	−1.59267
$448$	4.67211	0.220736
$449$	5.43278	0.256389	0.128194	−	0.991749i	$-0.459082\pi$
$449$	5.43278	0.256389	0.128194	+	0.991749i	$0.459082\pi$
$450$	0	0
$451$	−0.333209	−0.0156902
$452$	4.36184	0.205164
$453$	21.7320	1.02106
$454$	−11.2744	−0.529132
$455$	0	0
$456$	−9.52749	−0.446166
$457$	7.54789	0.353076	0.176538	−	0.984294i	$-0.443510\pi$
$457$	7.54789	0.353076	0.176538	+	0.984294i	$0.443510\pi$
$458$	−9.40494	−0.439464
$459$	−20.1960	−0.942670
$460$	0	0
$461$	−35.4323	−1.65025	−0.825124	−	0.564952i	$-0.808895\pi$
$461$	−35.4323	−1.65025	−0.825124	+	0.564952i	$0.808895\pi$
$462$	0.322987	0.0150267
$463$	20.9124	0.971883	0.485942	−	0.873991i	$-0.338477\pi$
$463$	20.9124	0.971883	0.485942	+	0.873991i	$0.338477\pi$
$464$	−4.20386	−0.195159
$465$	0	0
$466$	−3.22867	−0.149565
$467$	6.62523	0.306579	0.153289	−	0.988181i	$-0.451013\pi$
$467$	6.62523	0.306579	0.153289	+	0.988181i	$0.451013\pi$
$468$	−3.44204	−0.159109
$469$	57.8180	2.66979
$470$	0	0
$471$	−30.5303	−1.40676
$472$	13.5275	0.622653
$473$	−0.250387	−0.0115128
$474$	−13.3645	−0.613852
$475$	0	0
$476$	16.8598	0.772769
$477$	−3.98923	−0.182654
$478$	−17.8612	−0.816952
$479$	−29.4581	−1.34597	−0.672987	−	0.739655i	$-0.734989\pi$
$479$	−29.4581	−1.34597	−0.672987	+	0.739655i	$0.734989\pi$
$480$	0	0
$481$	5.26514	0.240070
$482$	26.6840	1.21542
$483$	−15.7560	−0.716923
$484$	−10.9980	−0.499907
$485$	0	0
$486$	6.66286	0.302233
$487$	8.80654	0.399063	0.199531	−	0.979891i	$-0.436058\pi$
$487$	8.80654	0.399063	0.199531	+	0.979891i	$0.436058\pi$
$488$	−4.27299	−0.193429
$489$	13.2510	0.599229
$490$	0	0
$491$	5.14258	0.232082	0.116041	−	0.993244i	$-0.462980\pi$
$491$	5.14258	0.232082	0.116041	+	0.993244i	$0.462980\pi$
$492$	−11.3089	−0.509844
$493$	−15.1701	−0.683227
$494$	−32.7492	−1.47345
$495$	0	0
$496$	−3.01051	−0.135176
$497$	−20.9943	−0.941723
$498$	−1.34829	−0.0604185
$499$	−12.9692	−0.580579	−0.290290	−	0.956939i	$-0.593752\pi$
$499$	−12.9692	−0.580579	−0.290290	+	0.956939i	$0.593752\pi$
$500$	0	0
$501$	−18.3259	−0.818739
$502$	−30.1552	−1.34589
$503$	17.2091	0.767314	0.383657	−	0.923476i	$-0.374664\pi$
$503$	17.2091	0.767314	0.383657	+	0.923476i	$0.374664\pi$
$504$	−3.05436	−0.136052
$505$	0	0
$506$	−0.0993641	−0.00441727
$507$	22.5499	1.00148
$508$	17.6572	0.783413
$509$	−12.1085	−0.536698	−0.268349	−	0.963322i	$-0.586478\pi$
$509$	−12.1085	−0.536698	−0.268349	+	0.963322i	$0.586478\pi$
$510$	0	0
$511$	32.8828	1.45465
$512$	1.00000	0.0441942
$513$	34.8110	1.53694
$514$	−3.15376	−0.139107
$515$	0	0
$516$	−8.49798	−0.374103
$517$	−0.193197	−0.00849681
$518$	4.67211	0.205281
$519$	−8.17201	−0.358711
$520$	0	0
$521$	−18.7936	−0.823362	−0.411681	−	0.911328i	$-0.635058\pi$
$521$	−18.7936	−0.823362	−0.411681	+	0.911328i	$0.635058\pi$
$522$	2.74824	0.120287
$523$	−42.9985	−1.88019	−0.940096	−	0.340909i	$-0.889265\pi$
$523$	−42.9985	−1.88019	−0.940096	+	0.340909i	$0.889265\pi$
$524$	−10.3103	−0.450406
$525$	0	0
$526$	14.0332	0.611875
$527$	−10.8638	−0.473233
$528$	0.0691309	0.00300853
$529$	−18.1528	−0.789252
$530$	0	0
$531$	−8.84350	−0.383775
$532$	−29.0605	−1.25993
$533$	−38.8725	−1.68375
$534$	15.2808	0.661263
$535$	0	0
$536$	12.3751	0.534525
$537$	−8.08619	−0.348945
$538$	−30.1257	−1.29881
$539$	0.669244	0.0288264
$540$	0	0
$541$	30.1751	1.29733	0.648664	−	0.761075i	$-0.275328\pi$
$541$	30.1751	1.29733	0.648664	+	0.761075i	$0.275328\pi$
$542$	14.8802	0.639161
$543$	27.1805	1.16643
$544$	3.60861	0.154718
$545$	0	0
$546$	37.6800	1.61255
$547$	−31.6736	−1.35426	−0.677132	−	0.735862i	$-0.736777\pi$
$547$	−31.6736	−1.35426	−0.677132	+	0.735862i	$0.736777\pi$
$548$	3.21863	0.137493
$549$	2.79344	0.119221
$550$	0	0
$551$	26.1480	1.11394
$552$	−3.37235	−0.143537
$553$	−40.7641	−1.73347
$554$	0.147278	0.00625724
$555$	0	0
$556$	20.9400	0.888055
$557$	−0.708971	−0.0300401	−0.0150200	−	0.999887i	$-0.504781\pi$
$557$	−0.708971	−0.0300401	−0.0150200	+	0.999887i	$0.504781\pi$
$558$	1.96810	0.0833163
$559$	−29.2104	−1.23547
$560$	0	0
$561$	0.249466	0.0105325
$562$	−25.5247	−1.07670
$563$	−29.9399	−1.26182	−0.630908	−	0.775857i	$-0.717318\pi$
$563$	−29.9399	−1.26182	−0.630908	+	0.775857i	$0.717318\pi$
$564$	−6.55699	−0.276099
$565$	0	0
$566$	0.0562691	0.00236517
$567$	−30.8892	−1.29722
$568$	−4.49354	−0.188545
$569$	21.1453	0.886456	0.443228	−	0.896409i	$-0.353833\pi$
$569$	21.1453	0.886456	0.443228	+	0.896409i	$0.353833\pi$
$570$	0	0
$571$	−19.5326	−0.817416	−0.408708	−	0.912665i	$-0.634020\pi$
$571$	−19.5326	−0.817416	−0.408708	+	0.912665i	$0.634020\pi$
$572$	0.237626	0.00993564
$573$	−4.77368	−0.199423
$574$	−34.4942	−1.43976
$575$	0	0
$576$	−0.653743	−0.0272393
$577$	1.08303	0.0450873	0.0225436	−	0.999746i	$-0.492824\pi$
$577$	1.08303	0.0450873	0.0225436	+	0.999746i	$0.492824\pi$
$578$	−3.97793	−0.165460
$579$	−13.4530	−0.559088
$580$	0	0
$581$	−4.11254	−0.170617
$582$	0.368245	0.0152642
$583$	0.275402	0.0114060
$584$	7.03811	0.291239
$585$	0	0
$586$	13.7033	0.566079
$587$	1.16839	0.0482244	0.0241122	−	0.999709i	$-0.492324\pi$
$587$	1.16839	0.0482244	0.0241122	+	0.999709i	$0.492324\pi$
$588$	22.7137	0.936698
$589$	18.7254	0.771566
$590$	0	0
$591$	10.7265	0.441230
$592$	1.00000	0.0410997
$593$	−6.01305	−0.246926	−0.123463	−	0.992349i	$-0.539400\pi$
$593$	−6.01305	−0.246926	−0.123463	+	0.992349i	$0.539400\pi$
$594$	−0.252586	−0.0103637
$595$	0	0
$596$	−21.9832	−0.900467
$597$	26.5303	1.08581
$598$	−11.5919	−0.474028
$599$	−11.9208	−0.487072	−0.243536	−	0.969892i	$-0.578307\pi$
$599$	−11.9208	−0.487072	−0.243536	+	0.969892i	$0.578307\pi$
$600$	0	0
$601$	21.7772	0.888309	0.444155	−	0.895950i	$-0.353504\pi$
$601$	21.7772	0.888309	0.444155	+	0.895950i	$0.353504\pi$
$602$	−25.9204	−1.05644
$603$	−8.09015	−0.329457
$604$	14.1877	0.577290
$605$	0	0
$606$	−8.60499	−0.349554
$607$	5.16486	0.209635	0.104818	−	0.994491i	$-0.466574\pi$
$607$	5.16486	0.209635	0.104818	+	0.994491i	$0.466574\pi$
$608$	−6.22000	−0.252254
$609$	−30.0849	−1.21910
$610$	0	0
$611$	−22.5386	−0.911813
$612$	−2.35910	−0.0953611
$613$	−39.5817	−1.59869	−0.799344	−	0.600873i	$-0.794820\pi$
$613$	−39.5817	−1.59869	−0.799344	+	0.600873i	$0.794820\pi$
$614$	−17.5823	−0.709565
$615$	0	0
$616$	0.210861	0.00849585
$617$	44.9917	1.81130	0.905650	−	0.424027i	$-0.139384\pi$
$617$	44.9917	1.81130	0.905650	+	0.424027i	$0.139384\pi$
$618$	−19.2484	−0.774283
$619$	30.0465	1.20767	0.603836	−	0.797108i	$-0.293638\pi$
$619$	30.0465	1.20767	0.603836	+	0.797108i	$0.293638\pi$
$620$	0	0
$621$	12.3217	0.494453
$622$	0.353133	0.0141594
$623$	46.6091	1.86735
$624$	8.06487	0.322853
$625$	0	0
$626$	−18.8717	−0.754265
$627$	−0.429994	−0.0171723
$628$	−19.9316	−0.795359
$629$	3.60861	0.143885
$630$	0	0
$631$	5.01448	0.199623	0.0998116	−	0.995006i	$-0.468176\pi$
$631$	5.01448	0.199623	0.0998116	+	0.995006i	$0.468176\pi$
$632$	−8.72499	−0.347061
$633$	10.3767	0.412435
$634$	15.1979	0.603587
$635$	0	0
$636$	9.34696	0.370631
$637$	78.0747	3.09343
$638$	−0.189728	−0.00751142
$639$	2.93762	0.116210
$640$	0	0
$641$	18.9133	0.747031	0.373515	−	0.927624i	$-0.378152\pi$
$641$	18.9133	0.747031	0.373515	+	0.927624i	$0.378152\pi$
$642$	−27.4788	−1.08450
$643$	22.9344	0.904444	0.452222	−	0.891906i	$-0.350632\pi$
$643$	22.9344	0.904444	0.452222	+	0.891906i	$0.350632\pi$
$644$	−10.2863	−0.405336
$645$	0	0
$646$	−22.4456	−0.883109
$647$	10.4741	0.411779	0.205889	−	0.978575i	$-0.433991\pi$
$647$	10.4741	0.411779	0.205889	+	0.978575i	$0.433991\pi$
$648$	−6.61139	−0.259720
$649$	0.610522	0.0239651
$650$	0	0
$651$	−21.5447	−0.844405
$652$	8.65087	0.338794
$653$	−44.7251	−1.75023	−0.875115	−	0.483916i	$-0.839214\pi$
$653$	−44.7251	−1.75023	−0.875115	+	0.483916i	$0.839214\pi$
$654$	4.74745	0.185640
$655$	0	0
$656$	−7.38299	−0.288257
$657$	−4.60111	−0.179506
$658$	−20.0000	−0.779681
$659$	15.1056	0.588431	0.294215	−	0.955739i	$-0.404942\pi$
$659$	15.1056	0.588431	0.294215	+	0.955739i	$0.404942\pi$
$660$	0	0
$661$	34.9654	1.36000	0.679998	−	0.733214i	$-0.261981\pi$
$661$	34.9654	1.36000	0.679998	+	0.733214i	$0.261981\pi$
$662$	−23.0210	−0.894736
$663$	29.1030	1.13027
$664$	−0.880231	−0.0341596
$665$	0	0
$666$	−0.653743	−0.0253320
$667$	9.25537	0.358369
$668$	−11.9640	−0.462901
$669$	−15.7094	−0.607361
$670$	0	0
$671$	−0.192848	−0.00744483
$672$	7.15650	0.276068
$673$	35.2496	1.35877	0.679385	−	0.733782i	$-0.262246\pi$
$673$	35.2496	1.35877	0.679385	+	0.733782i	$0.262246\pi$
$674$	34.0776	1.31262
$675$	0	0
$676$	14.7216	0.566217
$677$	−23.2835	−0.894858	−0.447429	−	0.894320i	$-0.647660\pi$
$677$	−23.2835	−0.894858	−0.447429	+	0.894320i	$0.647660\pi$
$678$	6.68125	0.256592
$679$	1.12321	0.0431049
$680$	0	0
$681$	−17.2695	−0.661769
$682$	−0.135870	−0.00520274
$683$	−11.8131	−0.452017	−0.226008	−	0.974125i	$-0.572568\pi$
$683$	−11.8131	−0.452017	−0.226008	+	0.974125i	$0.572568\pi$
$684$	4.06628	0.155478
$685$	0	0
$686$	36.5761	1.39648
$687$	−14.4060	−0.549624
$688$	−5.54789	−0.211511
$689$	32.1286	1.22400
$690$	0	0
$691$	14.1858	0.539652	0.269826	−	0.962909i	$-0.413034\pi$
$691$	14.1858	0.539652	0.269826	+	0.962909i	$0.413034\pi$
$692$	−5.33508	−0.202809
$693$	−0.137849	−0.00523646
$694$	13.8449	0.525547
$695$	0	0
$696$	−6.43926	−0.244080
$697$	−26.6423	−1.00915
$698$	11.3765	0.430607
$699$	−4.94552	−0.187057
$700$	0	0
$701$	19.2184	0.725870	0.362935	−	0.931815i	$-0.381775\pi$
$701$	19.2184	0.725870	0.362935	+	0.931815i	$0.381775\pi$
$702$	−29.4670	−1.11216
$703$	−6.22000	−0.234592
$704$	0.0451320	0.00170097
$705$	0	0
$706$	19.2631	0.724976
$707$	−26.2468	−0.987111
$708$	20.7207	0.778733
$709$	4.52525	0.169949	0.0849746	−	0.996383i	$-0.472919\pi$
$709$	4.52525	0.169949	0.0849746	+	0.996383i	$0.472919\pi$
$710$	0	0
$711$	5.70390	0.213913
$712$	9.97602	0.373867
$713$	6.62805	0.248222
$714$	25.8250	0.966478
$715$	0	0
$716$	−5.27905	−0.197288
$717$	−27.3589	−1.02174
$718$	14.6645	0.547273
$719$	0.915477	0.0341415	0.0170708	−	0.999854i	$-0.494566\pi$
$719$	0.915477	0.0341415	0.0170708	+	0.999854i	$0.494566\pi$
$720$	0	0
$721$	−58.7110	−2.18651
$722$	19.6884	0.732728
$723$	40.8732	1.52009
$724$	17.7448	0.659478
$725$	0	0
$726$	−16.8461	−0.625218
$727$	−11.7429	−0.435521	−0.217760	−	0.976002i	$-0.569875\pi$
$727$	−11.7429	−0.435521	−0.217760	+	0.976002i	$0.569875\pi$
$728$	24.5993	0.911710
$729$	30.0400	1.11259
$730$	0	0
$731$	−20.0202	−0.740473
$732$	−6.54515	−0.241916
$733$	27.6776	1.02230	0.511148	−	0.859493i	$-0.329220\pi$
$733$	27.6776	1.02230	0.511148	+	0.859493i	$0.329220\pi$
$734$	0.868349	0.0320514
$735$	0	0
$736$	−2.20164	−0.0811534
$737$	0.558514	0.0205731
$738$	4.82658	0.177669
$739$	−6.52187	−0.239911	−0.119956	−	0.992779i	$-0.538275\pi$
$739$	−6.52187	−0.239911	−0.119956	+	0.992779i	$0.538275\pi$
$740$	0	0
$741$	−50.1635	−1.84280
$742$	28.5099	1.04663
$743$	43.0091	1.57785	0.788925	−	0.614489i	$-0.210638\pi$
$743$	43.0091	1.57785	0.788925	+	0.614489i	$0.210638\pi$
$744$	−4.61135	−0.169060
$745$	0	0
$746$	32.4555	1.18828
$747$	0.575445	0.0210544
$748$	0.162864	0.00595489
$749$	−83.8152	−3.06254
$750$	0	0
$751$	−29.7068	−1.08402	−0.542008	−	0.840373i	$-0.682336\pi$
$751$	−29.7068	−1.08402	−0.542008	+	0.840373i	$0.682336\pi$
$752$	−4.28072	−0.156102
$753$	−46.1902	−1.68327
$754$	−22.1339	−0.806069
$755$	0	0
$756$	−26.1480	−0.950994
$757$	32.7918	1.19184	0.595920	−	0.803044i	$-0.296788\pi$
$757$	32.7918	1.19184	0.595920	+	0.803044i	$0.296788\pi$
$758$	−30.2511	−1.09877
$759$	−0.152201	−0.00552455
$760$	0	0
$761$	22.9411	0.831616	0.415808	−	0.909452i	$-0.363499\pi$
$761$	22.9411	0.831616	0.415808	+	0.909452i	$0.363499\pi$
$762$	27.0465	0.979790
$763$	14.4806	0.524232
$764$	−3.11649	−0.112751
$765$	0	0
$766$	14.4935	0.523673
$767$	71.2241	2.57175
$768$	1.53175	0.0552723
$769$	10.8123	0.389902	0.194951	−	0.980813i	$-0.437545\pi$
$769$	10.8123	0.389902	0.194951	+	0.980813i	$0.437545\pi$
$770$	0	0
$771$	−4.83078	−0.173976
$772$	−8.78278	−0.316099
$773$	−37.9989	−1.36673	−0.683363	−	0.730079i	$-0.739483\pi$
$773$	−37.9989	−1.36673	−0.683363	+	0.730079i	$0.739483\pi$
$774$	3.62689	0.130366
$775$	0	0
$776$	0.240408	0.00863014
$777$	7.15650	0.256738
$778$	36.9033	1.32305
$779$	45.9222	1.64533
$780$	0	0
$781$	−0.202802	−0.00725683
$782$	−7.94485	−0.284107
$783$	23.5274	0.840801
$784$	14.8286	0.529593
$785$	0	0
$786$	−15.7927	−0.563309
$787$	31.7730	1.13258	0.566292	−	0.824205i	$-0.308377\pi$
$787$	31.7730	1.13258	0.566292	+	0.824205i	$0.308377\pi$
$788$	7.00278	0.249464
$789$	21.4953	0.765252
$790$	0	0
$791$	20.3790	0.724594
$792$	−0.0295047	−0.00104840
$793$	−22.4979	−0.798923
$794$	22.9365	0.813986
$795$	0	0
$796$	17.3202	0.613900
$797$	9.02551	0.319700	0.159850	−	0.987141i	$-0.448899\pi$
$797$	9.02551	0.319700	0.159850	+	0.987141i	$0.448899\pi$
$798$	−44.5135	−1.57576
$799$	−15.4475	−0.546492
$800$	0	0
$801$	−6.52175	−0.230435
$802$	27.4332	0.968698
$803$	0.317643	0.0112094
$804$	18.9556	0.668513
$805$	0	0
$806$	−15.8508	−0.558319
$807$	−46.1450	−1.62438
$808$	−5.61775	−0.197632
$809$	29.7376	1.04552	0.522759	−	0.852481i	$-0.324903\pi$
$809$	29.7376	1.04552	0.522759	+	0.852481i	$0.324903\pi$
$810$	0	0
$811$	15.7650	0.553585	0.276793	−	0.960930i	$-0.410729\pi$
$811$	15.7650	0.553585	0.276793	+	0.960930i	$0.410729\pi$
$812$	−19.6409	−0.689260
$813$	22.7928	0.799378
$814$	0.0451320	0.00158187
$815$	0	0
$816$	5.52749	0.193501
$817$	34.5079	1.20728
$818$	−21.3920	−0.747954
$819$	−16.0816	−0.561937
$820$	0	0
$821$	−4.27798	−0.149303	−0.0746513	−	0.997210i	$-0.523784\pi$
$821$	−4.27798	−0.149303	−0.0746513	+	0.997210i	$0.523784\pi$
$822$	4.93014	0.171958
$823$	−13.8985	−0.484471	−0.242236	−	0.970217i	$-0.577881\pi$
$823$	−13.8985	−0.484471	−0.242236	+	0.970217i	$0.577881\pi$
$824$	−12.5663	−0.437766
$825$	0	0
$826$	63.2019	2.19908
$827$	−30.4944	−1.06039	−0.530197	−	0.847875i	$-0.677882\pi$
$827$	−30.4944	−1.06039	−0.530197	+	0.847875i	$0.677882\pi$
$828$	1.43930	0.0500192
$829$	0.493020	0.0171233	0.00856165	−	0.999963i	$-0.497275\pi$
$829$	0.493020	0.0171233	0.00856165	+	0.999963i	$0.497275\pi$
$830$	0	0
$831$	0.225593	0.00782574
$832$	5.26514	0.182536
$833$	53.5107	1.85404
$834$	32.0749	1.11066
$835$	0	0
$836$	−0.280721	−0.00970894
$837$	16.8487	0.582376
$838$	23.0891	0.797598
$839$	−27.9308	−0.964278	−0.482139	−	0.876095i	$-0.660140\pi$
$839$	−27.9308	−0.964278	−0.482139	+	0.876095i	$0.660140\pi$
$840$	0	0
$841$	−11.3276	−0.390606
$842$	−1.83757	−0.0633268
$843$	−39.0975	−1.34659
$844$	6.77438	0.233184
$845$	0	0
$846$	2.79849	0.0962141
$847$	−51.3837	−1.76556
$848$	6.10215	0.209549
$849$	0.0861901	0.00295804
$850$	0	0
$851$	−2.20164	−0.0754711
$852$	−6.88297	−0.235807
$853$	31.4029	1.07521	0.537607	−	0.843195i	$-0.319328\pi$
$853$	31.4029	1.07521	0.537607	+	0.843195i	$0.319328\pi$
$854$	−19.9639	−0.683150
$855$	0	0
$856$	−17.9395	−0.613158
$857$	39.7806	1.35888	0.679441	−	0.733731i	$-0.262223\pi$
$857$	39.7806	1.35888	0.679441	+	0.733731i	$0.262223\pi$
$858$	0.363983	0.0124262
$859$	41.8673	1.42849	0.714247	−	0.699894i	$-0.246769\pi$
$859$	41.8673	1.42849	0.714247	+	0.699894i	$0.246769\pi$
$860$	0	0
$861$	−52.8364	−1.80066
$862$	−15.9796	−0.544267
$863$	−8.52387	−0.290156	−0.145078	−	0.989420i	$-0.546343\pi$
$863$	−8.52387	−0.290156	−0.145078	+	0.989420i	$0.546343\pi$
$864$	−5.59662	−0.190401
$865$	0	0
$866$	−25.2525	−0.858116
$867$	−6.09319	−0.206936
$868$	−14.0654	−0.477412
$869$	−0.393776	−0.0133579
$870$	0	0
$871$	65.1568	2.20775
$872$	3.09936	0.104958
$873$	−0.157165	−0.00531922
$874$	13.6942	0.463213
$875$	0	0
$876$	10.7806	0.364243
$877$	−37.1163	−1.25333	−0.626664	−	0.779289i	$-0.715580\pi$
$877$	−37.1163	−1.25333	−0.626664	+	0.779289i	$0.715580\pi$
$878$	−9.30792	−0.314127
$879$	20.9901	0.707977
$880$	0	0
$881$	29.8633	1.00612	0.503060	−	0.864252i	$-0.332208\pi$
$881$	29.8633	1.00612	0.503060	+	0.864252i	$0.332208\pi$
$882$	−9.69410	−0.326417
$883$	−9.98952	−0.336174	−0.168087	−	0.985772i	$-0.553759\pi$
$883$	−9.98952	−0.336174	−0.168087	+	0.985772i	$0.553759\pi$
$884$	18.9998	0.639033
$885$	0	0
$886$	−9.52445	−0.319980
$887$	23.2703	0.781340	0.390670	−	0.920531i	$-0.372243\pi$
$887$	23.2703	0.781340	0.390670	+	0.920531i	$0.372243\pi$
$888$	1.53175	0.0514022
$889$	82.4965	2.76684
$890$	0	0
$891$	−0.298385	−0.00999628
$892$	−10.2559	−0.343392
$893$	26.6261	0.891008
$894$	−33.6728	−1.12619
$895$	0	0
$896$	4.67211	0.156084
$897$	−17.7559	−0.592852
$898$	5.43278	0.181294
$899$	12.6558	0.422094
$900$	0	0
$901$	22.0203	0.733602
$902$	−0.333209	−0.0110946
$903$	−39.7035	−1.32125
$904$	4.36184	0.145073
$905$	0	0
$906$	21.7320	0.721998
$907$	−34.0053	−1.12913	−0.564564	−	0.825389i	$-0.690956\pi$
$907$	−34.0053	−1.12913	−0.564564	+	0.825389i	$0.690956\pi$
$908$	−11.2744	−0.374153
$909$	3.67256	0.121811
$910$	0	0
$911$	−29.2297	−0.968424	−0.484212	−	0.874951i	$-0.660894\pi$
$911$	−29.2297	−0.968424	−0.484212	+	0.874951i	$0.660894\pi$
$912$	−9.52749	−0.315487
$913$	−0.0397266	−0.00131476
$914$	7.54789	0.249662
$915$	0	0
$916$	−9.40494	−0.310748
$917$	−48.1707	−1.59074
$918$	−20.1960	−0.666568
$919$	43.0238	1.41923	0.709613	−	0.704592i	$-0.248870\pi$
$919$	43.0238	1.41923	0.709613	+	0.704592i	$0.248870\pi$
$920$	0	0
$921$	−26.9317	−0.887430
$922$	−35.4323	−1.16690
$923$	−23.6591	−0.778748
$924$	0.322987	0.0106255
$925$	0	0
$926$	20.9124	0.687225
$927$	8.21510	0.269819
$928$	−4.20386	−0.137998
$929$	10.7786	0.353635	0.176817	−	0.984244i	$-0.443420\pi$
$929$	10.7786	0.353635	0.176817	+	0.984244i	$0.443420\pi$
$930$	0	0
$931$	−92.2340	−3.02285
$932$	−3.22867	−0.105759
$933$	0.540912	0.0177087
$934$	6.62523	0.216784
$935$	0	0
$936$	−3.44204	−0.112507
$937$	36.5755	1.19487	0.597435	−	0.801918i	$-0.296187\pi$
$937$	36.5755	1.19487	0.597435	+	0.801918i	$0.296187\pi$
$938$	57.8180	1.88782
$939$	−28.9067	−0.943336
$940$	0	0
$941$	58.9579	1.92197	0.960986	−	0.276596i	$-0.0892064\pi$
$941$	58.9579	1.92197	0.960986	+	0.276596i	$0.0892064\pi$
$942$	−30.5303	−0.994730
$943$	16.2547	0.529325
$944$	13.5275	0.440282
$945$	0	0
$946$	−0.250387	−0.00814079
$947$	9.63990	0.313255	0.156627	−	0.987658i	$-0.449938\pi$
$947$	9.63990	0.313255	0.156627	+	0.987658i	$0.449938\pi$
$948$	−13.3645	−0.434059
$949$	37.0566	1.20291
$950$	0	0
$951$	23.2794	0.754887
$952$	16.8598	0.546430
$953$	31.4809	1.01977	0.509884	−	0.860243i	$-0.329688\pi$
$953$	31.4809	1.01977	0.509884	+	0.860243i	$0.329688\pi$
$954$	−3.98923	−0.129156
$955$	0	0
$956$	−17.8612	−0.577672
$957$	−0.290616	−0.00939430
$958$	−29.4581	−0.951747
$959$	15.0378	0.485596
$960$	0	0
$961$	−21.9368	−0.707639
$962$	5.26514	0.169755
$963$	11.7278	0.377923
$964$	26.6840	0.859434
$965$	0	0
$966$	−15.7560	−0.506941
$967$	25.5439	0.821438	0.410719	−	0.911762i	$-0.365278\pi$
$967$	25.5439	0.821438	0.410719	+	0.911762i	$0.365278\pi$
$968$	−10.9980	−0.353488
$969$	−34.3810	−1.10448
$970$	0	0
$971$	43.7224	1.40312	0.701559	−	0.712611i	$-0.252488\pi$
$971$	43.7224	1.40312	0.701559	+	0.712611i	$0.252488\pi$
$972$	6.66286	0.213711
$973$	97.8341	3.13642
$974$	8.80654	0.282180
$975$	0	0
$976$	−4.27299	−0.136775
$977$	−44.6598	−1.42879	−0.714396	−	0.699741i	$-0.753299\pi$
$977$	−44.6598	−1.42879	−0.714396	+	0.699741i	$0.753299\pi$
$978$	13.2510	0.423719
$979$	0.450237	0.0143896
$980$	0	0
$981$	−2.02619	−0.0646912
$982$	5.14258	0.164106
$983$	9.17840	0.292745	0.146373	−	0.989230i	$-0.453240\pi$
$983$	9.17840	0.292745	0.146373	+	0.989230i	$0.453240\pi$
$984$	−11.3089	−0.360514
$985$	0	0
$986$	−15.1701	−0.483114
$987$	−30.6350	−0.975123
$988$	−32.7492	−1.04189
$989$	12.2144	0.388397
$990$	0	0
$991$	16.1246	0.512216	0.256108	−	0.966648i	$-0.417560\pi$
$991$	16.1246	0.512216	0.256108	+	0.966648i	$0.417560\pi$
$992$	−3.01051	−0.0955839
$993$	−35.2624	−1.11902
$994$	−20.9943	−0.665899
$995$	0	0
$996$	−1.34829	−0.0427223
$997$	8.99979	0.285026	0.142513	−	0.989793i	$-0.454482\pi$
$997$	8.99979	0.285026	0.142513	+	0.989793i	$0.454482\pi$
$998$	−12.9692	−0.410532
$999$	−5.59662	−0.177069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.be.1.4		5
5.2	odd	4		370.2.b.d.149.7	yes	10
5.3	odd	4		370.2.b.d.149.4	✓	10
5.4	even	2		1850.2.a.bd.1.2		5
15.2	even	4		3330.2.d.p.1999.1		10
15.8	even	4		3330.2.d.p.1999.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.4	✓	10	5.3	odd	4
370.2.b.d.149.7	yes	10	5.2	odd	4
1850.2.a.bd.1.2		5	5.4	even	2
1850.2.a.be.1.4		5	1.1	even	1	trivial
3330.2.d.p.1999.1		10	15.2	even	4
3330.2.d.p.1999.6		10	15.8	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} +4.67211 q^{7} +1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} +1.53175 q^{12} +5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} +3.60861 q^{17} -0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} +0.0451320 q^{22} -2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} -5.59662 q^{27} +4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} +1.00000 q^{32} +0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} +1.00000 q^{37} -6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} +7.15650 q^{42} -5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} -4.28072 q^{47} +1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} +5.26514 q^{52} +6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} -9.52749 q^{57} -4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} -3.01051 q^{62} -3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} +12.3751 q^{67} +3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} -0.653743 q^{72} +7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} +0.210861 q^{77} +8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} -7.38299 q^{82} -0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} -6.43926 q^{87} +0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} -2.20164 q^{92} -4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} +0.240408 q^{97} +14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9	\( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 + 3 * q^11 - 6 * q^13 - q^14 + 5 * q^16 + 9 * q^17 + 3 * q^18 + 4 * q^19 + 16 * q^21 + 3 * q^22 + 6 * q^23 - 6 * q^26 - q^28 + 11 * q^29 + 23 * q^31 + 5 * q^32 + 20 * q^33 + 9 * q^34 + 3 * q^36 + 5 * q^37 + 4 * q^38 + 20 * q^39 - 7 * q^41 + 16 * q^42 - 17 * q^43 + 3 * q^44 + 6 * q^46 + 12 * q^47 + 30 * q^49 - 20 * q^51 - 6 * q^52 - 7 * q^53 - q^56 + 11 * q^58 + 20 * q^59 - 9 * q^61 + 23 * q^62 - 33 * q^63 + 5 * q^64 + 20 * q^66 + 12 * q^67 + 9 * q^68 + 16 * q^69 + 6 * q^71 + 3 * q^72 - 6 * q^73 + 5 * q^74 + 4 * q^76 - q^77 + 20 * q^78 + 20 * q^79 - 7 * q^81 - 7 * q^82 + 12 * q^83 + 16 * q^84 - 17 * q^86 - 34 * q^87 + 3 * q^88 + 12 * q^89 + 16 * q^91 + 6 * q^92 - 4 * q^93 + 12 * q^94 + 3 * q^97 + 30 * q^98 - 11 * q^99

Introduction

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